Simplicity and the interface of algebra and geometry
Andrew Arana, University of Illinois at Urbana-Champaign
03:45-04:15 Friday April 5
[slides, video]

In the seventeenth century, there was a striking broadening of geometrical methods to include algebraic methods. Descartes, who was chiefly responsible for this broadening, claimed that the new curves admitted by his geometry were just as simple as those studied by the ancients, and thus were equally legitimate for geometrical study. I will firstly consider in what ways simplicity was a criterion of geometricity for ancient and Cartesian geometry. I will then explain how algebraic methods posed a challenge to this way of delineating geometry, and how Descartes resolved this challenge. If time permits, I will also consider the repercussions of this limitation for contemporary geometry.